What is the history behind the power of police to stop and question suspicious persons?

This is a difficult question to answer because, from what I can find, it hasn't been well documented. Also, I have to put forth the idea that I'm a not a lawyer and perhaps someone who specializes in legal history could shed a little more light on things than I can. Still, I think I can give it a fair go.

It appears that there is no one point at which law enforcement officials, in history, were given the ability to question suspicious persons out on the street. They've always been doing that. The procedure seems to have been adopted in the United States as part of English "common law." This was the adopting of practices that were legally accepted as common and acceptable to England at the time of the American Revolution. This was probably the easiest way to formulate a basic code of laws that could then be adapted for the new nation. Common appears to give law enforcement officials, such as they were, a broad ability to question people.

The amount of leeway police officers have had in questioning individuals, historically, appears to be dependent on how much intrusion the society at large is willing to tolerate. It doesn't appear the question was really tested much until the second half of the 20th century. This period coincided with a lot of civil strife, and in many places, an increase in crime, and people became more concerned with making sure that police procedures were fair and equitable. In keeping with this, many of the cases brought forward regarding the issues originated in urban areas.

You're already, it appears from another question you've asked, aware of the Terry vs. Ohio case, though this related more to the frisking aspect. Many of the cases are handled at the state level. It doesn't appear that they often make it to the Federal Supreme Court.

States have made rulings on the issue that are generally consistent and give a good deal of discretion to those "on the street," upholding the ideas of common law. There are a lot of gray areas, though. For example, when the police stop and question a person on the street without having much reason for doing so. The important angle to this is whether the person being stopped feels that they are free to leave at any time, or whether they feel that they have been restricted from leaving. The police can ask questions as they feel prudent, but in general, if the person being asked the questions hasn't demonstrated any other suspicious behavior he or she may very well be free to refuse. It is when the casual questioning morphs into an official "detainment" that the law becomes more demanding.

In short, the ability of the police to question people on the street appears to go back into common law and reflects the trust we put in such individuals to maintain order, prosecute crime, and keep our neighborhoods safe. Modern law has adjusted but not eliminated this ability in order to ensure an individual's rights under the 4th amendment are protected.